Blow-up analysis concerning singular Trudinger-Moser inequalities in dimension two

TL;DR

This paper establishes a sharp form of the singular Trudinger-Moser inequality in two dimensions and identifies extremal functions using blow-up analysis, addressing challenges posed by singularities.

Contribution

It provides a sharper version of the inequality and characterizes extremal functions, advancing understanding of singular Trudinger-Moser inequalities in two dimensions.

Findings

Derived a sharp singular Trudinger-Moser inequality

Identified extremal functions for the inequality

Analyzed asymptotic behavior near blow-up points

Abstract

In this paper, we derive a sharp version of the singular Trudinger-Moser inequality, which was originally established by Adimurthi and Sandeep (Nonlinear Differ. Equ. Appl. 2007). Moreover, extremal functions for those singular Trudinger-Moser inequalities are also obtained. Our method is the blow-up analysis. Compared with our previous work (J. Differential Equations 2015), the essential difficulty caused by the presence of singularity is how to analyse the asymptotic behaviour of certain maximizing sequence near the blow-up point. We overcome this difficulty by combining two different classification theorems of Chen and Li (Duke Math. J. 1991; Duke Math. J. 1995) to get the desired bubble.